Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Cyclic extensions of $K(\sqrt {-1})/K$HTML articles powered by AMS MathViewer

by Jón Kr. Arason, Burton Fein, Murray Schacher and Jack Sonn
Trans. Amer. Math. Soc. 313 (1989), 843-851 Request permission

Abstract:

In this paper the height ${\text {ht}}(L/K)$ of a cyclic $2$-extension of a field $K$ of characteristic $\ne 2$ is studied. Here ${\text {ht}}(L/K) \geq n$ means that there is a cyclic extension $E$ of $K,E \supset L$, with $[E:L] = {2^n}$. Necessary and sufficient conditions are given for ${\text {ht}}(L/K) \geq n$ provided $K(\sqrt { - 1})$ contains a primitive ${2^n}$th root of unity. Primary emphasis is placed on the case $L = K(\sqrt { - 1})$. Suppose ${\text {ht}}(K(\sqrt { - 1})/K) \geq 1$. It is shown that ${\text {ht}}(K(\sqrt { - 1})/K) \geq 2$ and if $K$ is a number field then ${\text {ht}}(K(\sqrt { - 1})/K) \geq n$ for all $n$. For each $n \geq 2$ an example is given of a field $K$ such that ${\text {ht}}(K(\sqrt { - 1})/K) \geq n$ but ${\text {ht}}(K(\sqrt { - 1})/K) \ngeq n + 1$.
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